In how many ways can the letters of the word ‘PERMUTATIONS’ be arranged if each word starts with P and ends with S?

Given: We have 12 letters


To Find: Number of words formed with Letter of the word ‘PERMUTATIONS.’


The formula used: The number of permutations of n objects, where p1 objects are of one kind, p2 are of the second kind, ..., pk is of a kth kind and the rest if any, are of a different kind is


In the word ‘PERMUTATIONS’ we have 2 T’s.


We have to start the word with P and end it with S, hence the first and last position is occupied with P and S respectively.


As two positions are occupied the remaining 10 positions are to be filled with 10 letters in which we have 2 T’s.


NOTE:- Unless specified , assume that repetition is not allowed.


Let us represent the arrangement


P














S




Hence,


The first place is occupied by P = 1 way


The last place (12th) is occupied by S = 1 way


For the remaining 10 places:


Using the above formula


Where,


n=10


p1=2



Total number of ways are ways.


In 1814400 ways the letters of the word ‘PERMUTATIONS’ can be arranged if each word starts with P and ends with S.


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